In what follows, we give some well known lemma which is useful for the proof.
Lemma 1 (2).
Let (X, d) be a metric space, x ∊ X and A ∊ IX. For α ∊ [ 0,1], if p
α
(x, A)=0 and [ A]
α
is closed subset of X, then x
α
⊂A.
Now, we introduce the concept of β-admissible in sense of Mohammadi et al. [13] for fuzzy mapping.
Definition 7.
Let (X, d) be a metric space, β:X × X → [0, ∞), α ∊ [ 0,1] and T:X → W
α
(X). A mapping T is said to be β-admissible if for each x∈X and y∈ [ T x]
α
, with β(x, y)≥1, we have β(y, z)≥1 for all z∈ [ T y]
α
.
Next, we prove the existence of fuzzy fixed point theorem for some generalized type of contraction fuzzy mapping. For our result, we let Ѱ be the family of non-decreasing functions Ѱ:[0, ∞)→[0, ∞) such that for each t>0. It is easy to see that for ψ∈Ѱ, ψ(t)<t for all t>0 and ψ(0)=0.
Theorem 1.
Let (X, d) be a complete metric space, α ∈ [ 0,1] and T be fuzzy mapping from X to W
α
(X). Suppose that there exist ψ∈Ѱ and β:X×X→[0, ∞) such that
(1)
for all x, y∈X, where L≥ 0 and
If the following condition holds,
-
(i)
T is β-admissible,
-
(ii)
there exist x 0∈X and x 1∈ [ T x 0]
α
such that β(x 0, x 1)≥1,
-
(iii)
if {x
n
} is sequence in X such that β(x
n
, x n+1)≥1 and x
n
→u as n→∞, then β(x
n
, u)≥1,
-
(iv)
ψ is continuous,
then there exists x∈X such that x
α
is a fuzzy fixed point of T.
Proof.
For x0∈X and x1∈ [ T x0]
α
in condition (ii), we get β(x0, x1)≥1. Since [T x1]
α
is nonempty compact subset of X, there exists x2∈ [ T x1]
α
, such that
(2)
From (2) and the fact that β(x0, x1)≥1, we have
By the same argument, for x2∈X, we have [ T x2]
α
, which is a nonempty compact subset of X and then there exists x3∈ [ T x2]
α
such that
(3)
For x0∈X and x1∈ [ T x0]
α
with β(x0, x1)≥1, by definition of β-admissible, we get
From (1), (3) and (4), we have
By induction, we can construct a sequence {x
n
} in X such that, for each , x
n
∈ [ T xn−1]
α
with β(xn−1, x
n
)≥1 and
where
Hence,
(5)
for all . If there exists which , then from Lemma 1, we have , that is is a fuzzy fixed point of T. Therefore, we suppose that for each , p
α
(x
n
, T x
n
)>0 and thus d(xn−1, x
n
)>0 for all . So, if d(x
n
, xn+1)>d(xn−1, x
n
) for some , then from (5) and ψ(t)<t for t∈(0, ∞), we have
which is a contradiction. Therefore, we have
(6)
Next, we will show that {x
n
} is a Cauchy sequence in X. Since continuous function ψ is belong to Ѱ, there exist ε>0 and positive integer h=h(ε) such that
Let m>n>h. Using the triangular inequality, previous relation and (6), we have
This implies that {x
n
} is a Cauchy sequence in X. By completeness of X, there exists x∈X such that x
n
→x as n→∞.
Finally, we show that p
α
(x, T x)=0. Assume on the contrary that p
α
(x, T x)>0. By condition (iii), we have β(x
n
, x)≥1 for all . Now we have
Letting n→∞, it follows that
which is a contradiction. Therefore, we have p
α
(x, T x)=0. Hence, by Lemma 1, x
α
⊂T x. This complete the proof.
Next, we give some examples to support the validity of our result.
Example 2.
Let X= [ 0,1] and d:X×X→[0, ∞) as d(x, y)=|x−y| for all x, y∈X. Then (X, d) is a complete metric space. Let us define T:X→IX by
Let . We observe that
for all x∈X. Therefore, T is fuzzy mapping from X to W
α
(X).
Define β:X×X→[0, ∞) by β(x, y)=2 for all x, y∈X. Then it is easy to check that T is an β-admissible. For each x, y∈X, we get
where for all t>0 and L≥0. It is easy to see that conditions (ii) and (iii) in Theorem 1 hold. Therefore, all conditions of Theorem 1 hold. Thus, T has an α-fuzzy fixed point x∈X, that is, a point .
By using Remark 1, we get the following result.
Theorem 2.
Let (X, d) be a complete metric space, α∈ [ 0,1] and T be fuzzy mapping from X to W
α
(X). Suppose that there exist ψ∈Ѱ and β:X×X→[0, ∞) such that
(7)
for all x, y∈X, where L≥0 and
If the following condition holds,
-
(i)
T is β *-admissible,
-
(ii)
there exist x 0∈X and x 1∈ [ T x 0]
α
such that β(x 0, x 1)≥1,
-
(iii)
if {x
n
} is sequence in X such that β(x
n
, x n+1)≥1 and x
n
→u as n→∞, then β(x
n
, u)≥1,
-
(iv)
ψ is continuous,
then there exists x∈X such that x
α
is a fuzzy fixed point of T.
In Theorems 1 and 2, we take ψ(t)=θ t, where θ∈(0,1) then we have the following corollary which is a fuzzy extension of fixed point theorem given by Berinde [16].
Corollary 1.
Let (X, d) be a complete metric space, α∈ [0,1] and T be fuzzy mapping from X to W
α
(X). Suppose that there exists β:X×X→[0, ∞) such that
(8)
for all x, y∈X, where, θ∈(0,1), L≥0 and
If the following condition holds,
-
(i)
T is β-admissible (or β ?-admissible),
-
(ii)
there exist x 0∈X and x 1∈ [ T x 0]
α
such that β(x 0, x 1)≥1,
-
(iii)
if {x
n
} is sequence in X such that β(x
n
, x n+1)≥1 and x
n
→u as n→∞, then β(x
n
, u)≥1,
then there exists x∈X such that x
α
is a fuzzy fixed point of T.
If we take L=0 in Theorems 1 and 2 and Corollary 1, then we have the following corollaries:
Corollary 2.
Let (X, d) be a complete metric space, α∈ [ 0,1] and T be fuzzy mapping from X to W
α
(X). Suppose that there exist ψ∈Ѱ and β:X×X→[0, ∞) such that
(9)
for all x, y∈X and
If the following condition holds,
-
(i)
T is β-admissible (or β *-admissible),
-
(ii)
there exist x 0∈X and x 1∈ [ T x 0]
α
such that β(x 0, x 1)≥1,
-
(iii)
if {x
n
} is sequence in X such that β(x
n
, x n+1)≥1 and x
n
→u as n→∞, then β(x
n
, u)≥1,
-
(iv)
ψ is continuous,
then there exists x∈X such that x
α
is a fuzzy fixed point of T.
Corollary 3.
Let (X, d) be a complete metric space, α∈ [ 0,1] and T be fuzzy mapping from X to W
α
(X). Suppose that there exists β:X×X→[0, ∞) such that
(10)
for all x, y∈X, where θ∈(0,1). If the following condition holds,
-
(i)
T is β-admissible (or β *-admissible),
-
(ii)
there exist x 0∈X and x 1∈ [ T x 0]
α
such that β(x 0, x 1)≥1,
-
(iii)
if {x
n
} is sequence in X such that β(x
n
, x n+1)≥1 and x
n
→u as n→∞, then β(x
n
, u)≥1,
then there exists x∈X such that x
α
is a fuzzy fixed point of T.
If we set β(x, y)=1 for all x, y∈X in Theorem 1 or Theorem 2, we get the following result:
Corollary 4.
Let (X, d) be a complete metric space, α∈ [ 0,1] and T be fuzzy mapping from X to W
α
(X). Suppose that there exist ψ∈Ѱ such that
(11)
for all x, y∈X, where L≥0 and
Then there exists x∈X such that x
α
is a fuzzy fixed point of T.
If ψ(t)=θ t, where θ∈(0,1) and β(x, y)=1 for all x, y∈X in Theorem 1 or Theorem 2, then we have the following corollary.
Corollary 5.
Let (X, d) be a complete metric linear space, α∈ [ 0,1] and T be fuzzy mapping from X to W
α
(X) such that
(12)
for all x, y∈X, where θ∈(0,1). Then there exists x∈X such that x
α
is a fuzzy fixed point of T.